Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include<iostream>
#include<iomanip>
using namespace std;
long long n;
double a[3000], sum, dp[3000][3000];
int main(){
	cin >> n;
	for (int i=1;i<=n;i++) cin >> a[i];
	dp[0][0]=1;
	for (int i=1;i<=n;i++)
		for (int j=0;j<=n;j++)
			dp[i][j]=dp[i-1][j]*((double)1-a[i])+(j?dp[i-1][j-1]*a[i]:0);
	for (int i=n;i>=(n+1)/2;i--){
		sum+=dp[n][i];
	}
	cout <<fixed << setprecision(10)<< sum<<'\n';
}